Traditional arithmetic tolerancing simply adds all the tolerances in a tolerance stack-up at the extremes of the drawing tolerances to predict a "worst case" assembly variation. It is important to note that if parts are built within tolerance and the assembly was correctly analyzed, a worst case approach assures 100% good assemblies.
Statistical tolerancing takes advantage of the fact that assemblies rarely or never stack in a worst case manner, and accepts the possibility that a small percentage of assemblies will fail to meet tolerance requirements. Under this approach, the tolerances of the detail parts can be increased because it can be shown that the statistical chance of worst case tolerance accumulation is small. Analysis indicates that the economic advantage accruing from the use of statistical tolerancing and the larger detail tolerances they make possible exceeds the cost of reworking or even scrapping the few assemblies that fail to meet the tolerance requirements. When statistical tolerancing is used to develop drawing requirements, both the design calculations and part inspection plans are more involved so normally only critical dimensions will be statistically controlled.
An assembly method known as "determinant assembly" is an approach to the production of large flexible parts and assemblies, such as airplanes, that eliminates the use of most traditional "hard tooling." An example of "determinant assembly" used to make airplane fuselage panels and fuselages is disclosed in U.S. patent application Ser. No. 07/964,533, now U.S. Pat. No. 5,560,102 entitled "Panel and Fuselage Assembly" filed on Oct. 13, 1992, by Micale and Strand. Another example of "determinant assembly" used in the airplane industry, this time to make airplane wings, is disclosed in U.S. Provisional Application 60/013,986 entitled "Determinant Wing Assembly" filed on Mar. 22, 1996, by Munk and Strand. To ensure that the assemblies, designed using the determinant assembly method, can be assembled successfully, tolerances should be analyzed to insure that the specified drawing tolerances will be producible and will support the preferred manufacturing plan/assembly sequence. Typical tolerance stack-ups for airplane assemblies require that a statistical tolerance analysis be performed in order to predict good assemblies made with producible detail part tolerances.
The "population" of manufactured parts, as used herein, is a term used to describe sets of numbers or values, consisting of measurements or observations about those parts. Populations of parts and the measurements thereof are described herein by distributions of these values. Such a description is usually given in terms of a frequency distribution, a probability distribution, or a density function with values given by f(x). Two parameters used to describe a population are its mean .mu. and its standard deviation .sigma., wherein .sigma..sup.2 called the population variance. These parameters characterize the center or location of a population and the variation around the center. More specifically, these parameters are defined in terms of f(x) by ##EQU1## In the discrete case the population consists of many finite values and in the continuous case the population is so large that it is more conveniently represented by a continuum of values and the distribution of values is described by a density function f(x). If the population is normally distributed, part measurements will distribute and divide approximately in the proportions as shown in FIG. 1.
It is often impractical or uneconomical to observe a very large population in its entirety. Instead, one obtains a random sample and, based on an examination of this random sample, one infers characteristics of interest about the full population. The purpose of most statistical investigations is to generalize from information contained in random samples characteristics of the population from which such samples are drawn. For example, in making inferences about the population parameters .mu. and .sigma..sup.2 based on a random sample X.sub.1, . . . , X.sub.n one calculates the corresponding sample estimates, namely the sample mean ##EQU2## and the sample variance ##EQU3## Here the divisor n-1 in the definition of S.sup.2 is motivated by a technical concern of unbiasedness in the estimator S.sup.2. In large samples it matters little whether one divides by n or by n-1.
A basic assumption of the statistical tolerance analysis approach discussed herein is that features of the produced parts can be described with a normal distribution. The probability density function for a normal distribution is ##EQU4##
The total area under the normal curve from x=-.infin. to x=+.infin. is equal to one; the area under f(x) between any two points a and b (a.ltoreq.b) is the proportion of part features between a and b.
Since the normal probability density function cannot be integrated in closed form between any pair of limits, probabilities or proportions of part features between such limits are usually obtained from tables of the standard normal distribution with mean .mu.=0 and standard deviation .sigma.=1. This is done by way of the following standardization:
If X represents a random element from a normal population with mean .mu. and standard deviation .sigma., then the population proportion of such elements falling within a,b! is ##EQU5## where Z=(X-.mu.)/.sigma. is a random element from a standard normal distribution and .PHI.(z) denotes the tabulated area under the standard normal density to the left of Z, i.e., ##EQU6## with standard normal density ##EQU7##
The most common statistical analysis case that arises in design occurs when random elements from two or more populations are combined in some specified manner. Determinant assembly techniques are usually concerned with assembling parts whose tolerances stack linearly, i.e., EQU X.sub.assy =a.sub.1 X.sub.1 +a.sub.2 X.sub.2 +...+a.sub.n X.sub.n
usually with coefficients a.sub.i =1 or a.sub.i =-1, depending on the direction of action of the i.sup.th element in the tolerance chain. When random elements from two or more populations are combined in a linear fashion they form a new and derived population with mean and variance given by ##EQU8## the latter simplification arising when a.sub.i.sup.2 =1 for all i. The resultant standard deviation is the square root of .sigma..sup.2.sub.assy.
Typically, statistical tolerancing is based on several assumptions:
Variations in part dimensions have a normal distribution. PA1 Production process is in statistical control (all variations occur at random). PA1 Process spread is equal to plus or minus three standard deviations, 6.sigma.. For a normally distributed population, 99.73% of the production parts will be within the process spread. PA1 1. Worst Case (Arithmetic) PA1 2. Simulation Analysis PA1 3. Modified Root Sum Square (RSS)
Statistical Process Control (SPC) provides standardized techniques to monitor manufacturing processes and verify process control and capability. To determine if the process is "capable," it is necessary to develop methods to calculate whether the variation is too large or if the process mean has shifted too far from nominal.
Once the detail part specification limits have been established and the natural variability of the process has been determined, the capability ratio, Cp, can be calculated as follows; ##EQU9## where USL and LSL are the upper and lower specification limits. The Cp capability ratio assumes that the measurements are normally distributed, but does not take into account the centering of data relative to the target value. It is simply the ratio of tolerance requirements to process capability.
The process capability index, Cpk, is a standard measure of process capability over an extended period of time for a process exhibiting statistical control. Cpk is considered to be a reliable indicator of process performance, taking into account process variation and deviation from nominal, ##EQU10## Cpk can be calculated as follows; ##EQU11## To determine whether a process is in statistical control, enough measurements are needed to allow all potential sources or variation to be represented. For any given period of time, a process characteristic will be considered in statistical control if all the plotted points in that period of time fall inside the control limits (.+-.3.sigma. limits).
When the process is centered within the specification limits, then Cp=Cpk. The following table shows the percent process fallout for shifts in Cpk for various values of Cp. The table considers shifts in the process from the center of the specification limits. To reduce the number of defective detail parts, the process can be centered or the variability can be reduced, or both can be done.
______________________________________ Percent Process Fallout For Shifts in Cpk from Various Values of Cp Shift in Cpk (Cp - Cpk Cp 0.00 0.20 0.40 ______________________________________ .50 13.361 20.193 38.556 1.00 .270 .836 3.594 1.20 .0318 .1363 .8198 1.40 .0027 .0160 .1350 ______________________________________
Three approaches to tolerance analysis are available for use for determinant assembly:
Selection of part datums and tolerance stack-up are the same for all methods. How we treat the part variation in the analysis is different for each approach.
The worst case analysis approach is well understood. It is simply the arithmetic sum of all tolerance contributors in an assembly stack-up. It is a conservative approach, requiring no knowledge about the individual detail part variation distribution since theoretically all parts could be made at either specification limit and the assembly will be within tolerance every time. This is the simplest analysis and most desirable from a fabrication standpoint since no knowledge is required about the part variation. If the calculated worst case tolerances are producible and predict a good assembly, these tolerances should be used.
A number of tolerance analysis software programs using statistical simulation techniques are available to predict the amount of variation that can occur in an assembly due to specified design tolerances, tool tolerances, and manufacturing/assembly variation. Some programs can determine the major contributing factors of the predicted variation and their percentage of contribution.
Simulation begins with a mathematical model of the assembly. Often data from a computer aided design program is an input to the model. The model includes detail geometry, tolerance variations (design and process) and the assembly sequence. The model simulates the production of a specified number of assemblies. During the simulations, the dimensions on each of the parts being assembled and on the assembly fixtures are randomly varied within the tolerances and statistical distributions specified. Output characteristics of interest are measured on the assemblies and the results are analyzed statistically.
The statistical analysis performed will give the percent of production assemblies that will be out of specification. The simulation can then be used to determine the major items contributing to the variation. Problem corrections can be identified and incorporated into the model. Additional simulations can be run to determine the effectiveness of the solution.
Three dimensional simulation programs require trained operators and dedicated equipment. Use is primarily limited to complicated structure or areas highly subject to change which is more difficult to analyze using the other more simplified approaches.
The RSS method of tolerance analysis is based on the assumption that tolerances stack linearly. Traditionally, the total tolerance band is set to 6.sigma. of the detail process capability. Therefore, the tolerance band can be expressed in terms of the standard deviation. ##EQU12## In the previous discussion on normal distributions, it was noted that for a linear stack ##EQU13## resulting in the well known root sum square (RSS) or statistical tolerance stacking formula.
Use of the RSS method of tolerance analysis has been observed to optimistically establish wider detail part tolerances and underestimate assembly variation. Thus, there has long been a need for a process for establishing valid detail part dimensional tolerance limits that will enable accurate prediction of an economically acceptable degree of non-conformance of a large flexible assembly made from said parts, especially in a process that accounts for detail part mean shifts of limited amounts in the process value of interest.